Diophantine equations for integers.

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Ex) Find gcd(8633, 8051)8051 97 83 83 8633 97 8989
Theorem(3.2.2, p.105)For any nonzero <strong>integers</strong> a and b, thereexist <strong>integers</strong> x and y such thatgcd(a, b) ax + by.Here’s how you use the EuclideanAlgorithm to write gcd(8633, 8051) as alinear combination of 8633 and 8051.
Page 1 and 2: The Euclidean Algorithm andDiophant
Page 3 and 4: Use of the gcdReducing fractionsEx.
Page 5: Euclidean Algorithm(p. 102)To find
Page 9 and 10: • Solve each division problem, ex
Page 11 and 12: Ex) Now use the Euclidean Algorithm
Page 13 and 14: Theorem 1For any nonzero integers a
Page 15 and 16: Theorem 4If gcd(a,b)│c, then the
Page 17 and 18: Using Division and Euclidean Algori
Page 19 and 20: • Use the Euclidean Algorithm to
Page 21 and 22: Solve 172x + 20y 10004 2 172 (1 7
Page 23 and 24: Theorem 4If the linear Diophantine
Page 25 and 26: ExampleFind all positive solutions
Page 27: Find all positive integer solutions

Diophantine equations for integers.

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